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Current Question (ID: 16022)

Question:

$\text{A source of sound gives 5 beats per second when sounded with another source of frequency 100 per second.}$ $\text{The second harmonic of the source, together with a source of frequency 205 per second, gives 5 beats per second.}$ $\text{What is the frequency of the source?}$

Options:

1. $95 \ \text{Hz}$
2. $100 \ \text{Hz}$
3. $105 \ \text{Hz}$
4. $205 \ \text{Hz}$

Solution:

$\text{Hint: no. of beats} = |v_1 - v_2|$ $\text{Step: Find the beat frequency in 1}^{\text{st}} \text{ case.}$ $f_1 - f_2 = \pm 5$ $\Rightarrow f_1 = f_2 \pm 5$ $\Rightarrow f_1 = 100 \pm 5$ $\Rightarrow f_1 = 105 \ \text{or} \ 95$ $\text{Step 2: Find the beat frequency in 2}^{\text{nd}} \text{ case.}$ $2f_1 - f_3 = \pm 5$ $\Rightarrow 2f_1 - 205 = \pm 5$ $\Rightarrow 2f_1 = 210 \ \text{or} \ 200$ $\Rightarrow f_1 = 105 \ \text{or} \ 100$ $\text{The second harmonic produces 5 beats per second with a 205 Hz sound.}$ $\text{This means its frequency is either 200 Hz or 210 Hz.}$ $\text{Since the fundamental frequency is 105 Hz, the second harmonic is}$ $2 \times 105 = 210 \ \text{Hz.}$ $\text{Therefore, the frequency of the source is 105 Hz.}$ $\text{Hence, option (3) is the correct answer.}$

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Expected JSON Format:

{
  "question": "The mass of carbon present in 0.5 mole of $\\mathrm{K}_4[\\mathrm{Fe(CN)}_6]$ is:",
  "options": [
    {
      "id": 1,
      "text": "1.8 g"
    },
    {
      "id": 2,
      "text": "18 g"
    },
    {
      "id": 3,
      "text": "3.6 g"
    },
    {
      "id": 4,
      "text": "36 g"
    }
  ],
  "solution": "\\begin{align}\n&\\text{Hint: Mole concept}\\\\\n&1 \\text{ mole of } \\mathrm{K}_4[\\mathrm{Fe(CN)}_6] = 6 \\text{ moles of carbon atom}\\\\\n&0.5 \\text{ mole of } \\mathrm{K}_4[\\mathrm{Fe(CN)}_6] = 6 \\times 0.5 \\text{ mol} = 3 \\text{ mol}\\\\\n&1 \\text{ mol of carbon} = 12 \\text{ g}\\\\\n&3 \\text{ mol carbon} = 12 \\times 3 = 36 \\text{ g}\\\\\n&\\text{Hence, 36 g mass of carbon present in 0.5 mole of } \\mathrm{K}_4[\\mathrm{Fe(CN)}_6].\n\\end{align}",
  "correct_answer": 4
}